On Creating A More Generalized Trott's Constant

[Joseph S. Myers]

On Fri, 23 Dec 2005, Hans Havermann wrote:

> ... giving us 81 decimal digits of agreement but, again, no obvious
> methodology with which to get more. I would guess that the reason
> digit-extension bogs down (eventually) is because of the interplay of how
> these numbers are created with the inherent (in)accuracy afforded by each
> system of number representation.

My digit-extension method here is to treat a sequence of digits as 
representing two intervals: the interval of numbers whose decimal 
expansions start with those digits, and the interval of numbers with a 
continued fraction expansion starting with those digits (i.e. the interval 
between the last two continued fraction convergents).  If those intervals 
have nonempty intersection, then (a) we can write down a number in the 
intersection and a continued fraction expansion for that number, agreeing 
to the given number of terms, and (b) we can try each possible next digit 
as an extension, rejecting it if the resulting subintervals no longer 
overlap.  In fact my experiments always find a possible digit-extension if 
the intervals overlap, though there may be cases where there isn't such an 
extension.

Choosing always the largest possible digit at each stage yields a number

0.33090909090629090909090...

which suggests a pattern which should allow it to be shown that such a 
number does exist.  The continued fraction [0; 3, 45, 2] is 0.3309090... 
which is very close to the above number; if my calculations are correct 
then the above with the "90" recurring at the end is [0; 3, 45, 2, 
4722549, 1, 2, 6, 1, 1, 1, 23, 2, 1, 4].  Thus as a next approximation to 
the decimal we write

0.3309090909062 [90 repeated 524727 times] 6126111909052131 [90 recurring]

where 9*524727+6 = 4722549; this number is slightly below the number with 
the previous continued fraction, hence the choice of "31" at the end of 
the given decimal, because (based on the parity of the number of terms) 
such a number will have a continued fraction expansion [0; 3, 45, 2, 
4722549, 1, 2, 6, 1, 1, 1, 23, 2, 1, 3, 1, ...].  Now the next term in the 
continued fraction expansion of the last rational will be exponentially 
larger than 524727; and this should repeat to give a real with 
exponentially increasing sequences of "90".  (It may also be necessary to 
bound above the length of the sequences such as "6126111909052131" 
simultaneously with bounding below the length of the "90" sequences.)

It's quite possible there does exist a rational number with the required 
property - necessarily ending with something like ...a0b0c0d0... with 
alternate digits zero; I haven't searched for one.

-- 
Joseph S. Myers
jsm at polyomino.org.uk